A simplified system of equations describing the 2-D flow of fluid of uniform depth , with an imposed temperature difference , under gravity , with buoyancy , thermal diffusivity , and kinematic viscosity . The full equations are
(1) |
(2) |
(3) |
In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified
system, periodic solutions of the form
(4) |
(5) |
Lorenz included the following terms in his system of equations,
(6) | |||
(7) | |||
(8) |
(9) | |||
(10) | |||
(11) |
(12) | |||
(13) | |||
(14) |
The Critical Points at (0, 0, 0) correspond to no convection, and the
Critical Points at
(15) |
(16) |
(17) |
See also Butterfly Effect, Rössler Model
References
Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 27-31, 1988.
Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.''
Physica D 9, 189-208, 1983.
Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, 1983.
Lorenz, E. N. ``Deterministic Nonperiodic Flow.'' J. Atmos. Sci. 20, 130-141, 1963.
Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science.
New York: Springer-Verlag, pp. 697-708, 1992.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.
© 1996-9 Eric W. Weisstein